Formulas for angles, arcs and degrees

I am not sure I understand exactly what you are asking for. You say that you are trying to calculate something in blue, but there is a lot of different blue lines and areas in your two pictures, and the blue things are not the same in the two pictures. First suggestion would be to make it clear just what you are having trouble calculating. It might also be helpful to know if your difficulty is trigonometry (I don't know how to calculate that quantity by any method) or Excel (I know how to calculate the quantity by hand, but do not know the Excel functions to perform the same calculation).

Currently, if I change the (Horizon Distance) I get the correct distances I want. However, I would also like to have it where I could Input the (Arc/Curvature Length) and the (Viewpoint Elevation)
and everything else change accordingly.

If I understand what you are trying to do here, I think the easiest way to do this is to have separate copies of the calculation sheet for each desired set of inputs. I often find that the easiest way for me to deal with different input scenarios is to have a copy of the calculation that can be used for each input scenario. You can see how I have done this in my unit circle spreadsheet (http://www.excelforum.com/tips-and-t...han-excel.html ) where I have three different but almost identical spreadsheets. The only difference between the three is the input format for the angle (degrees, radians, or multiples of pi radians).

I'm reasonably certain this will be something we can help you with. How much of the help you need will be with the trig and how much will be the Excel functions? If you already know the trig formulas, we should be able to easily help you enter those formulas into Excel. If you need us to derive the trig formulas for you, that could be different.

It would probably help to label your pictures. I do not know what "tangent X" is referring to, nor is it obvious which is B or C and so on.

If you are struggling with the trig part of the problem, I would strongly suggest you spend some time with the unit circle (whether my spreadsheet or whatever source for a description of the trig ratios as they relate to the unit circle). As I indicated in my other thread, I found that understanding the unit circle was a key part of learning and understanding trig. For example, if I understand correctly, the "elevation of object" is the same quantity that Wikipedia labels "exsecant" on its diagram of the unit circle (which is sec(x)-r). Recognizing that should make deriving the formula for "elevation of object" (given angle and radius) fairly straightforward.

I also note that all of your angles are given in degrees. Because it is a common mistake, you need to recognize that Excel (like all other programming languages) uses angles in radians for its trig functions. If you want your inputs and outputs to be in degrees, recognize that each of your trig functions is going to need to convert between degrees and radians.

Do you just want the great circle distance between two points on a sphere, or something more complicated?

I tend to believe that I must first understand the math before working too hard on the programming. Did you understand how I derived the trig formula for the "elevation of object"? If you failed to understand how I connected your "elevation" to the exsecant and then to the secant, then some of the others may not be easy to understand either.
I would note that arc length is related to angle through the values for a full circle as a proportionality arc/circumference=angle/2pi. That should explain how to get the "degrees of line B" value from city to city distance.
The purple (2986.30) line is a tangent, as is the yellow (972.70) line. Once you can see those two lines as tangents, it should be easy to see how to calculate the length of the purple line from the angle calculated above. You should also be able to see that the radius+view point elevation is a secant, and can use those values to compute the angle from the inverse cosine function (ACOS()).

How are we doing so far?

I note that the error is about 10 ppm at worst. At that level, I would guess that there are some small differences in implementation or in precision level or similar between the two. I expect that both algorithms can be considered correct and that the error is buried in the minor differences in implementation. Further analysis would probably require one to know in intimate detail the algorithms used to compute the values in the correct output column and then compare those to the algorithms used in this spreadsheet, and identify the differences.